Lifting curves simply

TL;DR

This paper establishes linear lower bounds for the degree of covers needed to lift curves to simple curves on hyperbolic surfaces, with bounds independent of the hyperbolic structure and varying with punctures.

Contribution

It provides new linear lower bounds for lifting curves on hyperbolic surfaces, improving previous results and analyzing different cases based on punctures.

Findings

Linear lower bounds for $f_\rho(L)$ independent of hyperbolic structure

Asymptotically linear growth of $f_\rho$ for puncture-free surfaces

Exponential lower bounds for surfaces with punctures

Abstract

We provide linear lower bounds for $f_\rho(L)$, the smallest integer so that every curve on a fixed hyperbolic surface $(S,\rho)$ of length at most $L$ lifts to a simple curve on a cover of degree at most $f_\rho(L)$. This bound is independent of hyperbolic structure $\rho$, and improves on a recent bound of Gupta-Kapovich. When $(S,\rho)$ is without punctures, using work of Patel we conclude asymptotically linear growth of $f_\rho$. When $(S,\rho)$ has a puncture, we obtain exponential lower bounds for $f_\rho$.